Brownian Motion - Visual Signatures

We see that most real data do not share the properties of Brownian motion.
The comparisons are clearest when we view differences instead of the raw data.
Here are difference graphs for two examples of Brownian motion.
For comparison, below these are difference graphs for four years of EMC closing prices (left) and Dell closing prices (right).
For later reference we note that when the thickness of the plotting line is approximately equal to the spacing between successive values, the bulk of the difference plot merges into a band.
Further, for Brownian motion (top two graphs) we observe
  1. The band is of approximately constant width.
  2. The outliers are relatively small (this is called the short tails property).
  3. The outliers are approximately uniformly distributed.
How well do real data satisfy these conditions? At least for EMC and Dell, the match appears to be weak.

Return to Brownian Motion.